TauLib · API Book III

TauLib.BookIII.Doors.CriticalLine

TauLib.BookIII.Doors.CriticalLine

Critical Line Theorem, K5 Off-Diagonal Exclusion, τ-Effective RH Statement, and Primorial RH Verification Protocol.

Registry Cross-References

  • [III.T19] Critical Line Theorem – critical_line_check

  • [III.P10] K5 Off-Diagonal Exclusion – k5_exclusion_check

  • [III.D30] τ-Effective RH Statement – tau_effective_rh_check

  • [III.P11] Primorial RH Verification Protocol – rh_protocol_check

Mathematical Content

III.T19 (Critical Line): CONDITIONAL on O3: self-adjointness of H_L forces all eigenvalues real, which via the spectral correspondence forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½.

III.P10 (K5 Off-Diagonal Exclusion): K5 forbids off-diagonal coupling at the lemniscate crossing point. Off-critical-line zeros would require imaginary spectral coupling, which K5 forbids.

III.D30 (τ-Effective RH): Computable predicate: for each primorial depth k, the finite-cutoff operator has only real eigenvalues.

III.P11 (Primorial RH Verification Protocol): Six-step verification protocol at each primorial level.

Tau.BookIII.Doors.critical_line_check

source def Tau.BookIII.Doors.critical_line_check (k : Denotation.TauIdx) :Bool

[III.T19] Critical line check at level k: all spectral modes have real eigenvalues (= natural numbers) and the spectral correspondence maps them consistently. Combines self-adjointness + O3. Equations

  • Tau.BookIII.Doors.critical_line_check k = Tau.BookIII.Doors.critical_line_check.go k 0 (k + 1) Instances For

Tau.BookIII.Doors.critical_line_check.go

source@[irreducible]

**def Tau.BookIII.Doors.critical_line_check.go (k : Denotation.TauIdx)

(n fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.critical_line_multi_check

source def Tau.BookIII.Doors.critical_line_multi_check (db : Denotation.TauIdx) :Bool

[III.T19] Critical line at multiple depths. Equations

  • Tau.BookIII.Doors.critical_line_multi_check db = Tau.BookIII.Doors.critical_line_multi_check.go db 1 (db + 1) Instances For

Tau.BookIII.Doors.critical_line_multi_check.go

source@[irreducible]

**def Tau.BookIII.Doors.critical_line_multi_check.go (db : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.k5_exclusion_check

source def Tau.BookIII.Doors.k5_exclusion_check (bound db : Denotation.TauIdx) :Bool

[III.P10] K5 off-diagonal exclusion: at each primorial level k, the B-lobe and C-lobe eigenvalues have zero off-diagonal coupling. The crossing-point boundary conditions enforce real spectral flow. Equations

  • Tau.BookIII.Doors.k5_exclusion_check bound db = Tau.BookIII.Doors.k5_exclusion_check.go bound db 1 1 ((bound + 1) * (db + 1)) Instances For

Tau.BookIII.Doors.k5_exclusion_check.go

source@[irreducible]

**def Tau.BookIII.Doors.k5_exclusion_check.go (bound db : Denotation.TauIdx)

(n k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.tau_effective_rh_check

source def Tau.BookIII.Doors.tau_effective_rh_check (db : Denotation.TauIdx) :Bool

[III.D30] τ-Effective RH: for each primorial depth k, the finite-cutoff operator H_{≤k} has only real eigenvalues, and the finite zeta has the correct zero structure. A computable predicate. Equations

  • Tau.BookIII.Doors.tau_effective_rh_check db = Tau.BookIII.Doors.tau_effective_rh_check.go db 1 (db + 1) Instances For

Tau.BookIII.Doors.tau_effective_rh_check.go

source@[irreducible]

**def Tau.BookIII.Doors.tau_effective_rh_check.go (db : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.rh_protocol_check

source def Tau.BookIII.Doors.rh_protocol_check (db : Denotation.TauIdx) :Bool

[III.P11] Primorial RH verification protocol at depth k. Six steps: (i) compute Spec(H_{≤k}), (ii) verify eigenvalues real, (iii) verify zero locations, (iv) tower coherence, (v) CRT consistency, (vi) record certificate. Returns true if all steps pass. Equations

  • Tau.BookIII.Doors.rh_protocol_check db = Tau.BookIII.Doors.rh_protocol_check.go db 1 (db + 1) Instances For

Tau.BookIII.Doors.rh_protocol_check.go

source@[irreducible]

**def Tau.BookIII.Doors.rh_protocol_check.go (db : Denotation.TauIdx)

(k fuel : ℕ) :Bool**

Equations

  • One or more equations did not get rendered due to their size. Instances For

Tau.BookIII.Doors.critical_line_5

source theorem Tau.BookIII.Doors.critical_line_5 :critical_line_check 5 = true

Tau.BookIII.Doors.critical_line_10

source theorem Tau.BookIII.Doors.critical_line_10 :critical_line_check 10 = true

Tau.BookIII.Doors.critical_line_multi_5

source theorem Tau.BookIII.Doors.critical_line_multi_5 :critical_line_multi_check 5 = true

Tau.BookIII.Doors.k5_exclusion_10_3

source theorem Tau.BookIII.Doors.k5_exclusion_10_3 :k5_exclusion_check 10 3 = true

Tau.BookIII.Doors.tau_effective_rh_5

source theorem Tau.BookIII.Doors.tau_effective_rh_5 :tau_effective_rh_check 5 = true

Tau.BookIII.Doors.rh_protocol_4

source theorem Tau.BookIII.Doors.rh_protocol_4 :rh_protocol_check 4 = true

Tau.BookIII.Doors.critical_line_1

source theorem Tau.BookIII.Doors.critical_line_1 :critical_line_check 1 = true

[III.T19] Structural: critical line at depth 1 (only prime 2).

Tau.BookIII.Doors.k5_eigenvalue_1

source theorem Tau.BookIII.Doors.k5_eigenvalue_1 :lemniscate_eigenvalue 1 = 1

[III.P10] Structural: eigenvalue of first mode equals 1.

Tau.BookIII.Doors.tau_rh_1

source theorem Tau.BookIII.Doors.tau_rh_1 :tau_effective_rh_check 1 = true

[III.D30] Structural: τ-effective RH at depth 1.